Statistical Decision Theory Econ 2148, fall 2017 Statistical decision theory Maximilian Kasy Department of Economics, Harvard University 1 / 53
Statistical Decision Theory Takeaways for this part of class 1. A general framework to think about what makes a “good” estimator, test, etc. 2. How the foundations of statistics relate to those of microeconomic theory. 3. In what sense the set of Bayesian estimators contains most “reasonable” estimators. 2 / 53
Statistical Decision Theory Examples of decision problems ◮ Decide whether or not the hypothesis of no racial discrimination in job interviews is true ◮ Provide a forecast of the unemployment rate next month ◮ Provide an estimate of the returns to schooling ◮ Pick a portfolio of assets to invest in ◮ Decide whether to reduce class sizes for poor students ◮ Recommend a level for the top income tax rate 3 / 53
Statistical Decision Theory Agenda ◮ Basic definitions ◮ Optimality criteria ◮ Relationships between optimality criteria ◮ Analogies to microeconomics ◮ Two justifications of the Bayesian approach 4 / 53
Statistical Decision Theory Basic definitions Components of a general statistical decision problem ◮ Observed data X ◮ A statistical decision a ◮ A state of the world θ ◮ A loss function L ( a , θ ) (the negative of utility) ◮ A statistical model f ( X | θ ) ◮ A decision function a = δ ( X ) 5 / 53
Statistical Decision Theory Basic definitions How they relate ◮ underlying state of the world θ ⇒ distribution of the observation X . ◮ decision maker: observes X ⇒ picks a decision a ◮ her goal: pick a decision that minimizes loss L ( a , θ ) ( θ unknown state of the world) ◮ X is useful ⇔ reveals some information about θ ⇔ f ( X | θ ) does depend on θ . ◮ problem of statistical decision theory: find decision functions δ which “make loss small.” 6 / 53
Statistical Decision Theory Basic definitions Graphical illustration Figure: A general decision problem decision function a=δ(X) observed data decision X a statistical model X~f(x,θ) state of the world loss θ L(a,θ) 7 / 53
Statistical Decision Theory Basic definitions Examples ◮ investing in a portfolio of assets: ◮ X : past asset prices ◮ a : amount of each asset to hold ◮ θ : joint distribution of past and future asset prices ◮ L : minus expected utility of future income ◮ decide whether or not to reduce class size: ◮ X : data from project STAR experiment ◮ a : class size ◮ θ : distribution of student outcomes for different class sizes ◮ L : average of suitably scaled student outcomes, net of cost 8 / 53
Statistical Decision Theory Basic definitions Practice problem For each of the examples on slide 2, what are ◮ the data X , ◮ the possible actions a , ◮ the relevant states of the world θ , and ◮ reasonable choices of loss function L ? 9 / 53
Statistical Decision Theory Basic definitions Loss functions in estimation ◮ goal: find an a ◮ which is close to some function µ of θ . ◮ for instance: µ ( θ ) = E [ X ] ◮ loss is larger if the difference between our estimate and the true value is larger Some possible loss functions: 1. squared error loss, L ( a , θ ) = ( a − µ ( θ )) 2 2. absolute error loss, L ( a , θ ) = | a − µ ( θ ) | 10 / 53
Statistical Decision Theory Basic definitions Loss functions in testing ◮ goal: decide whether H 0 : θ ∈ Θ 0 is true ◮ decision a ∈ { 0 , 1 } (accept / reject) Possible loss function: if a = 1 , θ ∈ Θ 0 1 L ( a , θ ) = if a = 0 , θ / ∈ Θ 0 c 0 else. truth θ ∈ Θ 0 θ / ∈ Θ 0 decision a 0 0 c 1 1 0 11 / 53
Statistical Decision Theory Basic definitions Risk function R ( δ , θ ) = E θ [ L ( δ ( X ) , θ )] . ◮ expected loss of a decision function δ ◮ R is a function of the true state of the world θ . ◮ crucial intermediate object in evaluating a decision function ◮ small R ⇔ good δ ◮ δ might be good for some θ , bad for other θ . ◮ Decision theory deals with this trade-off. 12 / 53
Statistical Decision Theory Basic definitions Example: estimation of mean ◮ observe X ∼ N ( µ , 1 ) ◮ want to estimate µ ◮ L ( a , θ ) = ( a − µ ( θ )) 2 ◮ δ ( X ) = α + β · X Practice problem (Estimation of means) Find the risk function for this decision problem. 13 / 53
Statistical Decision Theory Basic definitions Variance / Bias trade-off Solution: R ( δ , µ ) = E [( δ ( X ) − µ ) 2 ] = Var ( δ ( X ))+ Bias ( δ ( X )) 2 = β 2 Var ( X )+( α + β E [ X ] − E [ X ]) 2 = β 2 +( α +( β − 1 ) µ ) 2 . ◮ equality 1 and 2: always true for squared error loss ◮ Choosing b (and a ) involves a trade-off of bias and variance, ◮ this trade-off depends on µ . 14 / 53
Statistical Decision Theory Optimality criteria Optimality criteria ◮ Ranking provided by the risk function is multidimensional: ◮ a ranking of performance between decision functions for every θ ◮ To get a global comparison of their performance, have to aggregate this ranking into a global ranking. ◮ preference relationship on space of risk functions ⇒ preference relationship on space of decision functions 15 / 53
Statistical Decision Theory Optimality criteria Illustrations for intuition ◮ Suppose θ can only take two values, ◮ ⇒ risk functions are points in a 2D-graph, ◮ each axis corresponds to R ( δ , θ ) for θ = θ 0 , θ 1 . R(.,θ 1 ) R(.,θ 0 ) 16 / 53
Statistical Decision Theory Optimality criteria Three approaches to get a global ranking 1. partial ordering : a decision function is better relative to another if it is better for every θ 2. complete ordering, weighted average : a decision function is better relative to another if a weighted average of risk across θ is lower weights ∼ prior distribution 3. complete ordering, worst case : a decision function is better relative to another if it is better under its worst-case scenario. 17 / 53
Statistical Decision Theory Optimality criteria Approach 1: Admissibility Dominance: δ is said to dominate another function δ ′ if R ( δ , θ ) ≤ R ( δ ′ , θ ) for all θ , and R ( δ , θ ) < R ( δ ′ , θ ) for at least one θ . Admissibility: decisions functions which are not dominated are called admissible, all other decision functions are inadmissible. 18 / 53
Statistical Decision Theory Optimality criteria Figure: Feasible and admissible risk functions R(.,θ 1 ) feasible admissible R(.,θ 0 ) 19 / 53
Statistical Decision Theory Optimality criteria ◮ admissibility ∼ “Pareto frontier” ◮ Dominance only generates a partial ordering of decision functions. ◮ in general: many different admissible decision functions. 20 / 53
Statistical Decision Theory Optimality criteria Practice problem ◮ you observe X i ∼ iid N ( µ , 1 ) , i = 1 ,..., n for n > 1 ◮ your goal is to estimate µ , with squared error loss ◮ consider the estimators 1. δ ( X ) = X 1 2. δ ( X ) = 1 n ∑ i X i ◮ can you show that one of them is inadmissible? 21 / 53
Statistical Decision Theory Optimality criteria Approach 2: Bayes optimality ◮ natural approach for economists: ◮ trade off risk across different θ ◮ by assigning weights π ( θ ) to each θ Integrated risk: � R ( δ , π ) = R ( δ , θ ) π ( θ ) d θ . 22 / 53
Statistical Decision Theory Optimality criteria Bayes decision function: minimizes integrated risk, δ ∗ = argmin R ( δ , π ) . δ ◮ Integrated risk ∼ linear indifference planes in space of risk functions ◮ prior ∼ normal vector for indifference planes 23 / 53
Statistical Decision Theory Optimality criteria Figure: Bayes optimality R(.,θ 1 ) R(δ*,.) π(θ) R(.,θ 0 ) 24 / 53
Statistical Decision Theory Optimality criteria Decision weights as prior probabilities � π ( θ ) d θ < ∞ ◮ suppose 0 < � π ( θ ) d θ = 1 (normalize) ◮ then wlog ◮ if additionally π ≥ 0 ◮ then π is called a prior distribution 25 / 53
Statistical Decision Theory Optimality criteria Posterior ◮ suppose π is a prior distribution ◮ posterior distribution: π ( θ | X ) = f ( X | θ ) π ( θ ) m ( X ) ◮ normalizing constant = prior likelihood of X � m ( X ) = f ( X | θ ) π ( θ ) d θ 26 / 53
Statistical Decision Theory Optimality criteria Practice problem ◮ you observe X ∼ N ( θ , 1 ) ◮ consider the prior θ ∼ N ( 0 , τ 2 ) ◮ calculate 1. m ( X ) 2. π ( θ | X ) 27 / 53
Statistical Decision Theory Optimality criteria Posterior expected loss � R ( δ , π | X ) := L ( δ ( X ) , θ ) π ( θ | X ) d θ Proposition Any Bayes decision function δ ∗ can be obtained by minimizing R ( δ , π | X ) through choice of δ ( X ) for every X . Practice problem Show that this is true. Hint: show first that � R ( δ , π ) = R ( δ ( X ) , π | X ) m ( X ) dX . 28 / 53
Statistical Decision Theory Optimality criteria Bayes estimator with quadratic loss ◮ assume quadratic loss, L ( a , θ ) = ( a − µ ( θ )) 2 ◮ posterior expected loss: R ( δ , π | X ) = E θ | X [ L ( δ ( X ) , θ ) | X ] ( δ ( X ) − µ ( θ )) 2 | X � � = E θ | X = Var ( µ ( θ ) | X )+( δ ( X ) − E [ µ ( θ ) | X ]) 2 ◮ Bayes estimator minimizes posterior expected loss ⇒ δ ∗ ( X ) = E [ µ ( θ ) | X ] . 29 / 53
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